Blackjack progressive jackpot odds

Hi there. I work at a casino where theres a progressive blackjack jackpot.

To win the full jackpot you need to get 4 aces of the same colour as your first four cards dealt to one box.

I sat down one day and worked out (on paper) what i thought your chances of hitting the jackpot was. I worked it out as 1 in 391,190.83. with a calculator, i work it out as being about the same.

I live in the UK, so its UK 6-deck blackjack. We are assuming no card counting method is used, and the game is dealth from a fresh shoe. (or the used cards are unknown). There is no shuffling machine, dealt straight form a shoe.

I worked it out by saying:

first card is 1/13 because you can be dealt any ace.
then 11/311,
then 10/310,
then 9/309

i timesed these fractions together to get my answer.


My question come from this point onwards. have i calculated it correctly? A friend of mine disbeleives me. He thinks that it can be influenced by other factors such as the dealers up-card, the other players cards, how many other people are playing, the fact that some aces are behind the cut card ETC.

I have tried explainging he is WRONG by telling him its the same thing as calculating the very first 4 cards of the deck are aces of the same colour. and just because we need them in a different place in the shoe to give us the four aces to hit the jackpot it doesnt change it.

Ive tried simplifying things but its like talknig to a brick wall. Please reply and tell me im correct. (truthfully OFC) because this is bugging me to hell.

Quote: pingu

Hi there. I work at a casino where theres a progressive blackjack jackpot.

To win the full jackpot you need to get 4 aces of the same colour as your first four cards dealt to one box.

I sat down one day and worked out (on paper) what i thought your chances of hitting the jackpot was. I worked it out as 1 in 391,190.83. with a calculator, i work it out as being about the same.

I live in the UK, so its UK 6-deck blackjack. We are assuming no card counting method is used, and the game is dealth from a fresh shoe. (or the used cards are unknown). There is no shuffling machine, dealt straight form a shoe.

I worked it out by saying:

first card is 1/13 because you can be dealt any ace.
then 11/311,
then 10/310,
then 9/309

i timesed these fractions together to get my answer.


My question come from this point onwards. have i calculated it correctly? A friend of mine disbeleives me. He thinks that it can be influenced by other factors such as the dealers up-card, the other players cards, how many other people are playing, the fact that some aces are behind the cut card ETC.

I have tried explainging he is WRONG by telling him its the same thing as calculating the very first 4 cards of the deck are aces of the same colour. and just because we need them in a different place in the shoe to give us the four aces to hit the jackpot it doesnt change it.

Ive tried simplifying things but its like talknig to a brick wall. Please reply and tell me im correct. (truthfully OFC) because this is bugging me to hell.

You are correct. There was a similar question last week about how other cards change the probability, but they don't. Yes, the player to your right might get the Ace you need, making your hitting 4 Aces harder, but he could also NOT get it, increasing your odds by removing a "bad" card. The reason the odds are what they are is because they account for all of the possibilities.

Now, if you KNOW the dealer's upcard is an Ace or other players have revealed their cards, you can calculate new odds as more information is available. But before that, everything is the same. Just because the dealer is taking a card or some cards will never be seen because of the cut does not have an effect the odds.

I agree with your math if you really don't know any of the other cards (no counting). I don't think the cut card matters. Unseen cards are unseen cards whether they're before the cut card or not. But I think your friend is partly correct that the other cards that are dealt face up affect the math. In most cases, youll have at least a little bit of extra information, even if you aren't actively counting cards. Say you and I are playing alone at a table. Your first two cards are two black aces, and my first two cards are two 5s, and the dealer is showing a 7. Your odds of hitting two more black aces are now (10/307)*(9/306), because the dealer and I have removed some of the non-aces from the shoe.

Im sure that the postion of the cutting card does have an influance, you esentially CUTTING OUT part of the deck meaning that cards behind the the cutting card are no longer avalable to players, changing your math.....

Quote: bidhendy123

Im sure that the postion of the cutting card does have an influance, you esentially CUTTING OUT part of the deck meaning that cards behind the the cutting card are no longer avalable to players, changing your math.....

But since you don't have the information of whats behind the cut, it doesn't change your probability. Yes, all of the aces could be behind the cut, making the odds zero, but NONE of the Aces could be behind the cut, making your odds increase greatly. Since you don't know and can never know when the game starts, the odds remain the same. If you are going to keep track of what comes out and adjust your math as you go, that's a different question.

Think about it like this, a simplified example. You have a new, shuffled deck of cards. Without looking at them, you take 8 cards and destroy them. There are now 44 cards. What are the odds the first card you draw will be an Ace? It's still 4 in 52 (1 in 13) even though there are only 44 cards in front of you, the odds are still the same, because you don't have the information about which cards were destroyed. Or do the same thing and destroy 51 cards. There's one card left. The odds are still 1 in 13 it's an Ace.

thankyou cclub.

btw, bidhendy is the guy who disagrees with me.

ive tried explaining it like you did to him, i had a single deck, put a cut in it, and asked him ho many positions there are that the ace of spades might be in. he agreed 52. i asked him how many players were playing in a high card type game, he said 5, so i dealt 5 face down cards, and asked him what are the odds the third player has the ace of spades. he agreed that that was 1/52. but with the blackjack jackpot thing, he disagrees with me. Its like talking to a brick wall and i really want to just show him it simply doesnt matter.

as for your reply rd4, im talking about when you place the jackpot sidebet. when you place the sidebet, you dont know the dealers card. or any other cards. etc. so if you do watch an ace come out, oh well.

All your really betting is that there are 4 aces o fthe same colour in the right position in the deck. if your playing the dealer headsup, then you want the aces in positions 1,3,4 and 5. the people at the casino who agree with me agree that the way i worked it out would be sorrect if i was working out the top 4 aces, but they think it changes if you give the dealer an upcard etc.

Make this question a poll so bidhendy sees no support for his position.

i was hoping to get a reply from the wizard. he told me that if the wizard backs me up he would eat his words.

I went a bit further and said i would eat my rucksak if the wizard disagreed with me. lol.

i dont know how or if im able to make this a poll sorry.

I agree that the probability is 1 in 391191. The way I would express it is 2*COMBIN(12,4)/COMBIN(312,4).

This is going to be true for any four cards in the shoe, assuming no other information is known. If other information is known, then it would affect the probability.

This is similar to a question I sometimes get about card counting. Some new counters get very stressed if the dealer won't show them the burn card, because they won't know where to start a running count. Some will incorrectly assume it was a good card, and start a running count at -1. I tell them that one burn card is no big deal, and mathematically equivalent to putting it behind the cut card. In other words, to be perfect, one should start a running count at 0, and add 1/52 to the number of remaining decks for purposes of the true count conversion. For practical purposes, just ignore the burn card, if you can't see it.

It just goes to show that an unseen card is an unseen card, it doesn't matter where it is or why you didn't see it.

Quote: Wizard

I agree that the probability is 1 in 391191. The way I would express it is 2*COMBIN(12,4)/COMBIN(312,4).

This is going to be true for any four cards in the shoe, assuming no other information is known. If other information is known, then it would affect the probability.

This is similar to a question I sometimes get about card counting. Some new counters get very stressed if the dealer won't show them the burn card, because they won't know where to start a running count. Some will incorrectly assume it was a good card, and start a running count at -1. I tell them that one burn card is no big deal, and mathematically equivalent to putting it behind the cut card. In other words, to be perfect, one should start a running count at 0, and add 1/52 to the number of remaining decks for purposes of the true count conversion. For practical purposes, just ignore the burn card, if you can't see it.

It just goes to show that an unseen card is an unseen card, it doesn't matter where it is or why you didn't see it.

I was just about to post that the Wizard basically backed up the argument, though for a separate game, here:

https://wizardofvegas.com/forum/gambling/tables/2176-in-caribbean-stud-does-the-of-players-have-any-even-an-infinitesimal-affect-on-the-houses-edge/

But he's here to speak for himself!

Not bad odds for a sidebet. I've had four aces dealt to me at least twice, and I've seen them dealt to other people numerous times (even five aces once!). Can't remember the colors. What does the jackpot pay?

its progressive. at the moment its on about £90,000. (costs £0.50 to play the jackpot).

there are other payouts, 4 aces any colour is 10% of the main jackpot (currently about £9,000), 3 aces coloured is £500, 3 aces is £100 i think, and 2 aces coloured is £25. (all amounts below 4 aces are fixed amounts).

Wow, seems like a great pay structure. Anybody figure out the house edge?

Quote: pingu

its progressive. at the moment its on about £90,000. (costs £0.50 to play the jackpot).

there are other payouts, 4 aces any colour is 10% of the main jackpot (currently about £9,000), 3 aces coloured is £500, 3 aces is £100 i think, and 2 aces coloured is £25. (all amounts below 4 aces are fixed amounts).

(J)(1/13)(11/311)(10/310)(9/309)+(J/10)((1/13)(23/311)(22/310)(21/309)-(1/13)(11/311)(10/310)(9/309))+(500)(1/13)(11/311)(10/310)+(100)((1/13)(23/311)(22/310)-(1/13)(11/311)(10/310))+(25)(1/13)(11/311)-.5(12/13)-.5(1/13)(288/311)-.5(1/13)(12/311)(288/310)(287/309)=0

i believe this is the breakeven point. not sure if i did it correctly though. the wizard or miplet will probably need to verify my math.

if you have 4aces any color, it could end up as 4aces the same color, so i took that out so as to not repeat the same combinations. i did the same thing with 3aces any color and 3aces the same color. as for when you lose, i did if you get a nonace as the first card, an ace as the first card but a nonace as the 2nd card, an ace for the first card and an ace of a different color for the 2nd card then no ace for the 3rd or 4th cards.

if my equation is correct, the breakeven point is a jackpot amount of 70,363 or greater. at a jackpot of ~$90,000 the player advantage is ~10%.

sounds good. I always wondered at what point it became profitable to play. I just didnt know how to go about working i out.

I had a great idea for a blackjack progressive jackpot they could use instead, which would make the casino more money in the longrun.

I thought that if they had a jackpot that paid out on hitting 21 with many cards, lets say hitting 21 on 5 cards wins a [portion of jackpot, and 6 card 21 wins more etc. maybe 7 card 21 as well.

this way, people will be playing bad blackjack (taking a card on 4 card 17 for the chance to hit the jackpot etc) so the house is taking a much larger edge on the actual blackjack. and as the jackpot is funded by other players, theyre not losing out when that is hit. considering the jackpot is enough, it would be the right thing to do to take a card on 4 card 17 etc as well. i reckon this would make a great table game for the casino, as players will be funding something that makes people play in a way that makes for amuch larger house edge.

hitting a 17 versus an A if dealer hits on soft 17 isnt that terrible a play. add in some kind of bonus and its an easy hit.

Dealer Player Stand Hit

A 10,7 -0.513758 -0.578940
10 10,7 -0.418457 -0.580066
9 10,7 -0.422164 -0.549404
8 10,7 -0.384233 -0.500967

https://wizardofodds.com/blackjack/appendix9-6dh17r4.html

well i was just using that as an example, but you woud also have people hitting on much worse than 17. imagine getting a six card 20.... the jackpot for 7 card 21 is huge and if big enough would make you take a card on 20.

Quote: pingu

sounds good. I always wondered at what point it became profitable to play. I just didnt know how to go about working i out.

I had a great idea for a blackjack progressive jackpot they could use instead, which would make the casino more money in the longrun.

I thought that if they had a jackpot that paid out on hitting 21 with many cards, lets say hitting 21 on 5 cards wins a [portion of jackpot, and 6 card 21 wins more etc. maybe 7 card 21 as well.

this way, people will be playing bad blackjack (taking a card on 4 card 17 for the chance to hit the jackpot etc) so the house is taking a much larger edge on the actual blackjack. and as the jackpot is funded by other players, theyre not losing out when that is hit. considering the jackpot is enough, it would be the right thing to do to take a card on 4 card 17 etc as well. i reckon this would make a great table game for the casino, as players will be funding something that makes people play in a way that makes for amuch larger house edge.

I'm suprised you don't see more promotions like this because this is actually a really good idea for the casino.

yup. having players, to fund other players making them make bad moves they wouldnt normally, so the casino takes more money off the blackjack per dollar bet, plus theyre taking money off the jackpot (take a cut) and it gives people a chance for a big win. which is always tempting. :)